A025610 -id:A025610 - cp's OEIS frontend

A307162 a(n) is the smallest k such that A319100(k) = A025610(n).

1, 3, 8, 7, 24, 21, 120, 56, 1320, 63, 168, 22440, 252, 840, 516120, 504, 9240, 819, 14967480, 2184, 157080, 3276, 613666680, 10920, 3612840, 6552, 28842333960, 120120, 15561, 104772360, 32760, 1528643699880, 2042040, 62244, 4295666760, 207480, 90189978292920, 46966920, 124488

Offset: 1

Jianing Song, Mar 27 2019

nonn

A025610 is the range of A319100.
Let b = A319100. Note that:
- if k is an odd number, then b(2*k) = b(k), b(4*k) = 2*b(k), b(2^e*k) = 4*b(k) for e >= 3;
- if k is not divisible by 3, then b(3*k) = 2*b(k), b(3^e*k) = 6*b(k) for e >= 2;
- for all primes p > 3, if k is not divisible by p, then b(p^e*k) = b(p*k).
As a result, it is easy to see that for every n, a(n) is not congruent to 2 modulo 4 and is not divisible by 16 or 27 or p^2 for any prime p > 3.

Jianing Song, Table of n, a(n) for n = 1..1000

Cf. A025610, A319100, A002476, A007528, A121940, A057130.

isA025610(n) = omega(6*n)==2&&valuation(n,2)>=valuation(n,3)
b(n) = if(isA025610(n), i=1; while(A319100(i)!=n, i++); i)
for(n=1, 216, if(isA025610(n), print1(b(n), ", "))) \\ See A319100 for its program

p(j) = my(t=0,v=vector(j)); for(k=1, oo, if(prime(k)%6==1, t++; v[t]=prime(k)); if(t==j, return(v)))
q(i) = my(t=0,v=vector(i)); for(k=1, oo, if(prime(k)%6==5, t++; v[t]=prime(k)); if(t==i, return(v)))
b(i,j) = {
if(j<=1 && i<=2, my(M=[1,3,8;7,21,56]); return(M[j+1,i+1]));
if(j==0 && i>=3, my(Q=q(i-3)); return(24*prod(k=1, i-3, Q[k])));
if(j>=2 && i<=2, my(P=p(j-1), w=[9,36,72]); return(w[i+1]*prod(k=1, j-1, P[k])));
if(j>=1 && i>=3, my(P=p(j), Q=q(i-2)); return(prod(k=1, j-1, P[k])*8*prod(k=1, i-3, Q[k])*min(9*Q[i-2], 3*P[j])));
}
list(lim) = my(v=A025610(lim), u=vector(#v)); for(k=1, #v, my(i=valuation(v[k],2)-valuation(v[k],3), j=valuation(v[k],3)); u[k]=b(i,j)); u \\ Jianing Song, Jun 04 2019, See A025610 for its program

Let p(j) = A002476(j), q(i) = A007528(i), P(j) = Product_{k=1..j} p(k) = A121940(j) if j > 0, Q(i) = Product_{k=1..i} q(k) = A057130(i) if i > 0. If A025610(n) = 2^i*6^j, then:
(a) if i = 0, then a(n) = 1 if j = 0, 7 if j = 1 and 9*P(j-1) if j >= 2;
(b) if i = 1, then a(n) = 3 if j = 0, 21 if j = 1 and 36*P(j-1) if j >= 2;
(c) if i = 2, then a(n) = 8 if j = 0, 56 if j = 1 and 72*P(j-1) if j >= 2;
(d) if i >= 3, then a(n) = 24*Q(i-3) if j = 0 and P(j-1)*8*Q(i-3)*min{9*q(i-2), 3*p(j)} if j >= 1. [Rewritten by Jianing Song, Jun 04 2019]

A343329 a(n) is the largest positive integer that is abundant and has the same prime signature as A025610(n) or 0 if no such number exists.

Original entry on oeis.org

0, 0, 0, 0, 0, 20, 0, 104, 0, 196, 464, 0, 1372, 1952, 0, 15376, 7232, 17576, 0, 119072, 32128, 476656, 0, 1032256, 130304, 7263392, 0, 8064128, 14776336, 522752, 131096512, 0, 66324736, 458066416, 2087936, 2024096128, 0, 533729792, 16649257024, 8382464, 33759290624, 27027081632, 0

Offset: 1

David A. Corneth, Apr 12 2021

nonn

a(n) = 0 if A025487(n) is a power of 2 or A025487(n) = 6.

			a(6) = 20 as A025610(6) = 12 and has prime signature (1, 2) as it is 2^(2) * 3^(1). The largest abundant number with prime signature (1, 2) is 20 = 2^(2) * 5^(1).

Cf. A005101, A025610.

A025629 Numbers of form 6^i*10^j with i, j >= 0.

Original entry on oeis.org

1, 6, 10, 36, 60, 100, 216, 360, 600, 1000, 1296, 2160, 3600, 6000, 7776, 10000, 12960, 21600, 36000, 46656, 60000, 77760, 100000, 129600, 216000, 279936, 360000, 466560, 600000, 777600, 1000000, 1296000, 1679616, 2160000, 2799360, 3600000, 4665600

Offset: 1

David W. Wilson

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A025610, A025618, A025622, A025626, A025627, A025628.

Cf. A025612, A025616, A025621, A025625, A025632, A025634.

n = 10^6; Flatten[Table[6^i*10^j, {i, 0, Log[6, n]}, {j, 0, Log10[n/6^i]}]] // Sort (* Amiram Eldar, Sep 26 2020 *)

list(lim)=my(v=List(), N); for(n=0, logint(lim\=1, 10), N=10^n; while(N<=lim, listput(v, N); N*=6)); Set(v) \\ Charles R Greathouse IV, Jan 10 2018

Sum_{n>=1} 1/a(n) = (6*10)/((6-1)*(10-1)) = 4/3. - Amiram Eldar, Sep 26 2020
a(n) ~ exp(sqrt(2*log(6)*log(10)*n)) / sqrt(60). - Vaclav Kotesovec, Sep 26 2020
a(n) = 6^A025663(n) * 10^A025688(n). - R. J. Mathar, Jul 06 2025

A025627 Numbers of form 6^i*8^j, with i, j >= 0.

Original entry on oeis.org

1, 6, 8, 36, 48, 64, 216, 288, 384, 512, 1296, 1728, 2304, 3072, 4096, 7776, 10368, 13824, 18432, 24576, 32768, 46656, 62208, 82944, 110592, 147456, 196608, 262144, 279936, 373248, 497664, 663552, 884736, 1179648, 1572864, 1679616, 2097152

Offset: 1

David W. Wilson

Amiram Eldar, Table of n, a(n) for n = 1..10000

Subsequence of A025610.

n = 10^6; Flatten[Table[6^i*8^j, {i, 0, Log[6, n]}, {j, 0, Log[8, n/6^i]}]] // Sort (* Amiram Eldar, Sep 26 2020 *)

Sum_{n>=1} 1/a(n) = (6*8)/((6-1)*(8-1)) = 48/35. - Amiram Eldar, Sep 26 2020
a(n) ~ exp(sqrt(2*log(6)*log(8)*n)) / sqrt(48). - Vaclav Kotesovec, Sep 26 2020
a(n) = 6^A025661(n) * 8^A025674(n). - R. J. Mathar, Jul 06 2025

A025626 Numbers of form 6^i*7^j, with i, j >= 0.

Original entry on oeis.org

1, 6, 7, 36, 42, 49, 216, 252, 294, 343, 1296, 1512, 1764, 2058, 2401, 7776, 9072, 10584, 12348, 14406, 16807, 46656, 54432, 63504, 74088, 86436, 100842, 117649, 279936, 326592, 381024, 444528, 518616, 605052, 705894, 823543, 1679616, 1959552

Offset: 1

David W. Wilson

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A025610, A025622, A025627, A025628, A025629.

Cf. A025619, A025631, A025632, A003591, A003594, A003595, A036566.

n = 10^6; Flatten[Table[6^i*7^j, {i, 0, Log[6, n]}, {j, 0, Log[7, n/6^i]}]] // Sort (* Amiram Eldar, Sep 25 2020 *)

Sum_{n>=1} 1/a(n) = (6*7)/((6-1)*(7-1)) = 7/5. - Amiram Eldar, Sep 25 2020
a(n) ~ exp(sqrt(2*log(6)*log(7)*n)) / sqrt(42). - Vaclav Kotesovec, Sep 25 2020
a(n) = 6^A025660(n) * 7^A025668(n). - R. J. Mathar, Jul 06 2025

A166470 a(n) = 2^F(n+1)*3^F(n), where F(n) is the n-th Fibonacci number, A000045(n).

Original entry on oeis.org

2, 6, 12, 72, 864, 62208, 53747712, 3343537668096, 179707499645975396352, 600858794305667322270155425185792, 107978831564966913814384922944738457859243070439030784

Offset: 0

Matthew Vandermast, Nov 05 2009

G. C. Greubel, Table of n, a(n) for n = 0..16

Subsequence of A025610 and hence of A003586 and A025487.

Cf. A000045, A000301, A002110, A010098, A115033, A166469, A174666, A230900.

[2^Fibonacci(n+1)*3^Fibonacci(n): n in [0..14]]; // G. C. Greubel, Jul 29 2024

3^First[#] 2^Last[#]&/@Partition[Fibonacci[Range[0,12]],2,1] (* Harvey P. Dale, Aug 20 2012 *)

a(n)=2^fibonacci(n+1)*3^fibonacci(n) \\ Charles R Greathouse IV, Sep 19 2022

[2^fibonacci(n+1)*3^fibonacci(n) for n in range(15)] # G. C. Greubel, Jul 29 2024

a(n) = A000301(n+1)*A010098(n).
For n > 1, a(n) = a(n-1)*a(n-2).
For m > 1, n > 1, A166469(A002110(m)*(a(n)^k)/12) = k*Fibonacci(m+n).
A166469(a(n)) = Fibonacci(n+2) + 1 = A001611(n+2).
a(n) = 2 * A174666(n+1). - Alois P. Heinz, Sep 16 2022
a(n) = 2^(Fibonacci(n+1) + c*Fibonacci(n)), with c=log_2(3). Cf. A000301 (c=1) & A010098 (c=2). - Andrea Pinos, Sep 29 2022
a(n) = A115033(2*n+1). - David Radcliffe, May 31 2025

Typo corrected by Matthew Vandermast, Nov 07 2009

A025614 Numbers of form 3^i*6^j, with i, j >= 0.

Original entry on oeis.org

1, 3, 6, 9, 18, 27, 36, 54, 81, 108, 162, 216, 243, 324, 486, 648, 729, 972, 1296, 1458, 1944, 2187, 2916, 3888, 4374, 5832, 6561, 7776, 8748, 11664, 13122, 17496, 19683, 23328, 26244, 34992, 39366, 46656, 52488, 59049, 69984, 78732, 104976, 118098

Offset: 1

David W. Wilson

Amiram Eldar, Table of n, a(n) for n = 1..10000

A025628 is a subsequence.
Cf. A003586, A003593, A003594, A003597, A107364.
Cf. A025610, A025618, A025622, A025626, A025627.

n = 10^6; Flatten[Table[3^i*6^j, {i, 0, Log[3, n]}, {j, 0, Log[6, n/3^i]}]] // Sort (* Amiram Eldar, Sep 26 2020 *)

Sum_{n>=1} 1/a(n) = (3*6)/((3-1)*(6-1)) = 9/5. - Amiram Eldar, Sep 26 2020
a(n) ~ exp(sqrt(2*log(3)*log(6)*n)) / sqrt(18). - Vaclav Kotesovec, Sep 26 2020
a(n) = 3^A025641(n) *6^A025657(n). - R. J. Mathar, Jul 06 2025

A025636 Exponent of 2 (value of i) in n-th number of form 2^i*6^j.

Original entry on oeis.org

0, 1, 2, 0, 3, 1, 4, 2, 5, 0, 3, 6, 1, 4, 7, 2, 5, 0, 8, 3, 6, 1, 9, 4, 7, 2, 10, 5, 0, 8, 3, 11, 6, 1, 9, 4, 12, 7, 2, 10, 5, 0, 13, 8, 3, 11, 6, 1, 14, 9, 4, 12, 7, 2, 15, 10, 5, 0, 13, 8, 3, 16, 11, 6, 1, 14, 9, 4, 17, 12, 7, 2, 15, 10, 5, 18, 0, 13, 8, 3, 16, 11, 6, 19, 1, 14, 9, 4, 17, 12, 7, 20, 2

Offset: 1

David W. Wilson

nonn

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A025610.

with(priqueue):
initialize(pq):
insert([-1,0,0],pq):
R:= NULL:
for count from 1 to 100 do
  t:= extract(pq);
  R:= R, t[2];
  insert([t[1]*2,t[2]+1,t[3]],pq);
  if t[2] = 0 then insert([t[1]*6,0,t[3]+1],pq) fi;
od:
R; # Robert Israel, Sep 27 2024

A279537 Numbers of the form 2^i * 6^j * 30^k.

Original entry on oeis.org

1, 2, 4, 6, 8, 12, 16, 24, 30, 32, 36, 48, 60, 64, 72, 96, 120, 128, 144, 180, 192, 216, 240, 256, 288, 360, 384, 432, 480, 512, 576, 720, 768, 864, 900, 960, 1024, 1080, 1152, 1296, 1440, 1536, 1728, 1800, 1920

Offset: 1

Charles R Greathouse IV, Dec 14 2016

nonn

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Intersection of A025487 and A051037.

Cf. A025610.

mx = 6000; Take[ Sort[ Flatten[ Table[2^i*6^j*30^k, {i, 0, Log[2, mx]}, {j, 0, Log[6, mx]}, {k, 0, Log[30, mx]}]]], 60] (* Robert G. Wilson v, Dec 14 2016 *)

list(lim)=my(v=List(),x,y); for(k=0,logint(lim\=1,30), y=30^k; for(j=0,logint(lim\y,6), x=y*6^j; while(x<=lim, listput(v,x); x<<=1))); Set(v)

Sum_{n>=1} 1/a(n) = 72/29. - Amiram Eldar, Feb 18 2021

A025692 Index of 2^n within sequence of numbers of form 2^i*6^j.

Original entry on oeis.org

1, 2, 3, 5, 7, 9, 12, 15, 19, 23, 27, 32, 37, 43, 49, 55, 62, 69, 76, 84, 92, 101, 110, 119, 129, 139, 150, 161, 172, 184, 196, 208, 221, 234, 248, 262, 276, 291, 306, 322, 338, 354, 371, 388, 406, 424, 442, 461, 480, 499, 519, 539, 560, 581, 602, 624, 646, 669, 692, 715

Offset: 0

David W. Wilson

nonn

Positions of zeros in A025656. - R. J. Mathar, Jul 06 2025

Robert Israel, Table of n, a(n) for n = 0..2500

Cf. A000079, A025610.

N:= 100: # for a(0)..a(N)
S:= sort([seq(seq(2^i*6^j,i=0..ilog2(2^N/6^j)),j=0..floor(log[6](2^N)))]):
seq(ListTools:-BinarySearch(S,2^i),i=0..N); # Robert Israel, Jan 12 2021

Offset corrected by Robert Israel, Jan 12 2021

cp's OEIS Frontend

A307162 a(n) is the smallest k such that A319100(k) = A025610(n).

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A343329 a(n) is the largest positive integer that is abundant and has the same prime signature as A025610(n) or 0 if no such number exists.

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A025629 Numbers of form 6^i*10^j with i, j >= 0.

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A025627 Numbers of form 6^i*8^j, with i, j >= 0.

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A025626 Numbers of form 6^i*7^j, with i, j >= 0.

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A166470 a(n) = 2^F(n+1)*3^F(n), where F(n) is the n-th Fibonacci number, A000045(n).

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A025614 Numbers of form 3^i*6^j, with i, j >= 0.

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A025636 Exponent of 2 (value of i) in n-th number of form 2^i*6^j.

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A279537 Numbers of the form 2^i * 6^j * 30^k.

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